Massachusetts Institute of Technology
Sloan School of Management
Ug
Working Paper
Design and Analysis of a Smart Market
For Industrial Procurement
J&6rmie Gallien
Lawrence M. Wein
October 15, 2000
Working Paper Number 4137
Contact Address:
Prof. Lawrence M.Wein
Sloan School of Management
Massachusetts Institute of Technology
Cambridge, MA 02142-1343
lwein@mit.edu
http://web.mit.edu/lwein/www/
Design and Analysis of a Smart Market for Industrial
Procurement
Jr mie Gallien
Operations Research Center, MIT
Cambridge, MA 02139
e-mail: jgallienOmit.edu
Lawrence M. Wein
Sloan School of Management, MIT
Cambridge, MA 02139
e-mail: lweinOmit.edu
October 15, 2000
Abstract
This paper addresses the problem of designing multi-item procurement auctions in capacity-constrained
environments. Using insights from classical auction theory, we construct an optimization-based auction
mechanism ("smart market") relying on the dynamic resolution of a linear program minimidng the buyer's
cost under the suppliers' capacity constraints. Suppliers can modify their offers in response to the optimal
allocation corresponding to each set of bids, giving rise to a dynamic competitive bidding process. To assist
suppliers, we also develop a bidding suggestion device based on a myopic best response (MBR) calculation
that solves an associated optimization problem. Assuming linear costs for the suppliers, we study within
a game-theoretic framework the sequence of bids arising in this smart market. Under a weak behavioral
assumption and some symmetry requirements, an explicit upper bound for the winning bids is established.
We then formulate a complete behavioral model and solution methodology based on the MBR rationale,
and show that the bounds derived earlier continue to hold. We analytically derive some structural and
convergence properties of the MBR dynamics in the simplest non-trivial market-environment, which suggests
further possible design improvements, and investigate bidding dynamics and incentive compatibility issues
via numerical simulations. In particular, experiments suggest that suppliers can be relied upon to provide
truthful capacity information when procurement contracts are properly designed.
1. Introduction
1.1. Background and Motivation.
Online business-to-business auctions are becoming
increasingly widespread, because their demand revelation properties are superior to that
1
of fixed-price mechanisms, while the Internet keeps their transaction costs low: In a May
2000 study by Forrester Research, Inc. (Kafka et al. 2000), it was estimated that the total
annual volume of transactions realized through online business-to-business auctions in the
U.S. would reach $746 billion by 2004. However, online implementations of traditional open
auction mechanisms are often poorly adapted to the selection of industrial suppliers in procurement markets, because this process typically involves capacity constraints, transportation costs, incumbent supplier switching costs, complex quality requirements, etc.: Transfer
price, although the sole focus of these mechanisms, is only one consideration among several
when it comes to choosing suppliers. Consequently, an important challenge faced by online
procurement service providers is to design alternative auction mechanisms adapted to these
complex allocation environments.
An example of such a mechanism is the bidding process used by FreeMarkets (Rangan
1999). One of the emerging leaders in electronic industrial auctions, FreeMarkets organizes
"bidding events" for large industrial buyers typically seeking to auction off procurement
contracts for specified quantities of several component types (or lots). In the mechanism
currently used, the various component types are ordered, and each component is effectively
auctioned off in a sequential manner. More precisely, binding bids on selling prices are
solicited for each component type from competing suppliers, who can observe the value of
the lowest bid submitted thus far. When a pre-specified time limit is reached for the first
component type in the list (after perhaps some overtime period triggered by an activitybased criteria), bidding closes for that component type, competition then turns to the next
component type, and this process repeats itself until the list of component types required by
the buyer is exhausted. However, in contrast to traditional auctions, the actual allocation
2
of procurement contracts is only decided by the buyer after the entire bidding event is
completed, and the lowest bidder may not be awarded the contract for a component. The
reason for postponing the allocation decision is to allow the buyer to take into account some
of the non-price factors mentioned earlier.
Although this mechanism appears to represent current best practice as of the summer
of 2000 (it is also used by FreeMarkets' competitor eBreviate), several sources suggest directions for improvement. In particular, a consequence of the delayed allocation decision is
that suppliers' bids are based on very limited competitive information, which appears to be
especially damaging in environments characterized by supplier capacity constraints: During
our meeting with FreeMarkets co-founder Sam Kinney (Kinney 2000), he hypothesized that
some suppliers grew nervous during the bidding event about receiving a total allocation exceeding their production capacity, especially if they had submitted relatively low bids on one
or several of the components auctioned early on. As a result, these suppliers would withhold
their bids on later components, thereby hampering competition. Indeed, some buyers also
seem to be aware of this issue; for example, Jack Porter, a manager of central purchasing
direct at Caterpillar, made the following comment during an interview on his experience as a
FreeMarkets customer (Janhke 1998): "This concept only applies in those areas where there
is a great deal of capacity."
Motivated by these observations, our objective is to design and analyze a stylized online
multi-item procurement auction mechanism adapted to supply environments with production
capacity constraints. Following a review of the relevant literature, we describe in §2 the
market environment under study, the proposed auction mechanism, and the specifics of the
game formulation. In §3, we discuss our solution methodology, derive some convergence
3
bounds, and investigate bidding dynamics and incentive compatibility issues. Concluding
remarks are provided in §4.
1.2. Literature Review. The study of multi-item auctions has been one of the most active
research areas in microeconomics lately (see Klemperer 1999 for a recent survey). It is also
becoming an increasingly popular topic in operations management, operations research and
computer science. The crux of the problem lies in the interdependencies between the various
objects being auctioned off, which arise in practice from factors such as capacity constraints,
budget constraints, economies of scale/scope, and legal restrictions. In our model, capacity
constraints are the only source of interdependencies across component types, and there are
no interdependencies between different units of the same component type, since component
production costs are assumed to be linear. Our discussion of the literature (restricted to
papers studying multi-item auctions) is organized in two parts: The first is relevant to the
design of our auction mechanism, while the second addresses outcome analysis.
The design of appropriate multi-item auction mechanisms is the object of impassioned
debate, motivated in large part by the sales of broadband spectrum licenses by the Federal
Communication Commission that began in the early nineties. A pivotal issue is whether
bidders should be allowed to submit bids on packages or bundles of items (Bykowski et al.
1995), giving rise to so-called "combinatorial auctions" (Rothkopf et al. 1998, Kelly 1999,
de Vries and Vohra 2000) that rely on optimization algorithms to determine the winners, in
contrast with adaptations of more traditional auction formats where bids are only allowed
for individual items (Milgrom 1999). A widespread concern about combinatorial auctions is
the allocation complexity they induce (Rothkopf et al. 1998). While the market mechanism
we study in this paper is also an optimization-based auction, the allocation complexity it
4
addresses does not stem from package bidding, but rather from the production capacity
constraint of each supplier across the multiple component types auctioned off: It relies on a
simple linear program (LP), so that computational complexity is not an issue.
More generally, our auction format belongs to the family of market mechanisms known in
the literature as smart markets, which are exchange institutions supported by a computer
executing an optimization algorithm to solve the allocation problem associated with each
given set of bids. This approach has been applied to a variety of market environments (see
McCabe et al. 1991 for an early survey), including airport runway allocation (Rassenti et al.
1982), space shuttle/station payload allocation (Banks et al. 1989, Plott and Porter 1996),
railroad track allocation (Brewer and Plott 1996), natural gas pipeline networks (McCabe
et al. 1989), truckload allocation (Caplice 1996), project management (Ledyard et al. 1994,
Walsh and Wellman 1998), general assignment problems (Olson and Porter 1994) and distributed scheduling (Kutanoglu and Wu 2000, Wellman et al. 1999). To our knowledge,
Stanley et al. (1954) contains the earliest occurrence in the literature of using optimization
to compute an allocation in a competitive bidding process. It is also the smart-market paper
that is closest to our work from a contextual standpoint: They formulate a LP to determine
the optimal allocation of procurement contracts for a fixed set of (sealed) bids submitted by
army clothing manufacturers with capacity constraints. While the transportation problem
they suggest is almost identical to the LP we use as the allocation engine of our mechanism,
a distinguishing feature of our work is that we study analytically the dynamic competitive
process that arises as the real-time solution to this LP generates competitive information
feedback to the bidders, who can then modify their bids, thereby leading to a new instance
of the LP. To our knowledge, the only other studies investigating analytically the properties
5
of bidding sequences in an iterative optimization-based auction mechanism are Demange et
al. (1986), Wellmann et al. (1999) and Parkes and Ungar (2000), which are discussed further
below.
In designing the information structure of our mechanism, we found relevant the discussion by Cramton (1998) of how ascending auctions compare to sealed-bid auctions. More
specifically, we have tried to take advantage of the design flexibility allowed by organizing
the auction on a distributed network of computers such as the Internet in order to achieve
a satisfactory balance between facilitating competition and preventing collusion. An important issue related to the former objective, which we feel is overlooked in the vast majority
of the papers cited above, is how bidders confronted with a complex multi-item allocation
mechanism can be assisted in making their bidding decisions: Rothkopf et al. (1998) show
that the complexity of the bidder's "minimal winning bid" problem is tightly connected
with that of the allocation (or "winner determination") problem, so that a human bidder's
computational abilities are likely to be quickly overwhelmed in all smart markets of interest. Recognizing that this concern is not only relevant to bidders, but also to the design of
an efficient auction mechanism, we investigate a particular form of competitive information
feedback that is called myopic best response (MBR). This feature allows each supplier to
enter his production costs into a private calculation device that will compute on his behalf
the bids maximizing his potential payoff in the next round, under the assumption that his
competitors' bids remain unchanged (Fudenberg and Tirole 1996); Note that the dynamic
minimum ask price posting rules described in Demange et al. (1986) and Parkes and Ungar
(2000) also address the limited computational abilities of human bidders. However, these
rules require bidders to evaluate in real-time their utility over the set of all subsets of items
6
being traded. Moreover, if the quantities of items to be allocated are continuous decision
variables then these rules could, in theory, be adapted via discretization, but perhaps at
the expense of computational intractability. The novelty of our MBR feedback is to fully
account for the optimization problem that will be solved by the buyer in the next round
when performing this calculation for the bidder.
We now turn to the relevant literature on outcome prediction. Because the computational
requirements associated with the assumption of full rationality are particularly formidable in
this game under uncertainty (Chapter 8 in Fudenberg and Tirole 1996, Samuelson 1997), we
follow instead the approach developed in the theory of tatonnement and learning in games
(Fudenberg and Levine 1998, Samuelson 1997); our rationale for using this approach is discussed in more detail in §3.1. That is, we explore the implications of less-than-fully rational
behavioral models on the final outcome. Under a. "weak" behavioral assumption defined in
§3.2, some symmetry requirements, and a condition on the minimum level of competition,
we derive an upper bound on the winning bids at the end of the auction, which to our
knowledge is the first explicit outcome prediction as a function of the market environment
parameters in the smart-market literature. We also propose a complete behavioral model
(justified at some length in §3.3.1) that is a natural adaptation to our setting of the myopic
best response bidding strategies assumed in Parkes and Ungar (2000) in a generic combinatorial auction, also referred to as "straightforward" bidding by Demange et al. (1986) in an
assignment auction, by Wellman et al. (1999) in a decentralized scheduling mechanism and
by Milgrom (1999) in the simultaneous ascending auction. These authors establish explicit
relationships in their respective settings between the final bids and the minimum equilibrium
bids, and between the final allocation and the surplus-maximizing (i.e. efficient) allocation.
7
In contrast, we consider the traditional perspective of a revenue-maximizing auctioneer; Although revenue-maximizing auctions are efficient if there is a resale market after the auction
(Ausubel and Cramton, 1999), this is unlikely to occur in procurement settings. From the
revenue-maximizing viewpoint, we derive some upper bounds on the final bids, and hence
the buyer's final procurement cost, in terms of the specific data of the procurement problem
under study (production costs, capacity constraints), and investigate how they are affected
by the initial bids. We also provide a numerical study of some incentive compatibility issues,
mostly related to the suppliers' revelation of their capacity constraints.
2. Mechanism Description and Game Definition
2.1. Market Environment. The initial situation giving rise to our market design problem is a monopsony where a buyer (typically a large manufacturing corporation) wants to
purchase some specified quantity of m component types (namely qj units of each component
type j in {1, ... , m}). Facing this opportunity are n suppliers ready to compete on selling
prices for the award of the corresponding procurement contracts. We assume that the only
parameters left to specify in these contracts are the total quantity along with the corresponding selling price for each component type. In particular, the rules by which payment terms,
delivery schedules, insurance provisions, etc. will be determined are clearly defined from the
outset and accepted by suppliers. In addition to the buyer's quantity requirements and her
low cost objective, the other consideration relevant to the allocation of components is that
each supplier has a limited overall production capacity across all component types. In the
present study, we assume that the capacity limitation of each supplier i E {1, ... , n} can be
appropriately described by a linear model specifying a total amount c, of production resource
8
available (e.g., machine-hours or man-hours), and the amounts aij of this resource needed
to produce one unit of each component type j. We also assume these capacity constraints
to be exogenous, so that suppliers do not have the option of acquiring additional production
capacity as part of the competitive interaction under study (see Budde and Gox 1999 for an
analysis of a procurement auction with endogenous capacity decisions).
The remainder of this section is organized as follows: The core of the proposed optimizationbased allocation mechanism is presented in §2.2, along with some practical implementation
issues. Section 2.3 focuses on the suppliers' payoff function, which is subsequently used both
to derive outcome predictions in §3, and to generate more efficient competitive information
feedback. The design of this information feedback is discussed in §2.4, which more generally
summarizes the information structure underlying this competitive interaction.
2.2. Proposed Mechanism.
Bidding in our mechanism takes place through a discrete
set of bidding rounds, denoted by t E N (t =
is the initial round). More precisely, each
supplier i is invited in each round t of the auction to simultaneously bid on his unit selling
price bij(t) for all component types j in {1,... ,m}.
Each bid bij (t) corresponds to an offer in round t by supplier i to sell any quantity between
0 and qj of component type j at a unit price given by the value of the bid (see §2.3 for a
discussion of how this relates to the suppliers' objective). The first constraint on these
bidding decisions is a classical non-reneging rule (i.e. bij(t) < bij(t') for t' < t) imposed to
ensure the efficiency of this mechanism: Increasing a bid submitted in some previous round
would amount to reneging a prior offer to sell, and thus threaten the stability of the price
formation process. The second constraint is a common multiple rule (i.e., bij(t) E eN for some
e > O)that further restricts the set of admissible bids to a regular lattice with granularity e,
9
or "e-grid". This rule both enforces a minimum bid decrement frequently imposed in practice
(see Aeppel 1999), and prevents strategic collusion signals using the lower digits of the bids
(reported in Kelly and Steinberg 1999 for example). In addition, the common multiple rule
resolves a technical definition problem associated with the solution concept we introduce in
§3.3.
The key feature of our auction format is that the overall capacity constraint of each
supplier will always be satisfied by all possible allocation outcomes, which seems desirable
from both the suppliers' and the buyer's perspective, as discussed in §1. This is achieved in
two steps: First, the capacity constraints ((aij)jE1,,m}
Ci)
of each supplier i are estimated
before the beginning of the auction. The parameters of these capacity constraints can be
either assessed by an auditor during a pre-certification/qualification of supplier i prior to
the auction, or merely provided by supplier i. In the latter case, the incentive compatibility
issues linked to this capacity revelation become particularly important, and are discussed
in §3.3.4. Second, once all the bids have been submitted for round t during the auction,
the mechanism calculates the quantity allocation xij(t) of component type j to supplier i
minimizing the buyer's total procurement cost, subject to both the quantity requirement
constraint for each component type and the capacity constraint of each supplier (estimated
during the first step mentioned above). This allocation x(t) = (ij(t))
iE(l,...,n} is obtained
jby
solving
in
each
round
tthe
following
P,
referred
of
to
in
the
remainder
this paper as...m}
by solving in each round t the following LP, referred to in the remainder of this paper as
10
AE[b(t)] (for "allocation engine"):
m
Min E E bij(t)xij
Zi
/i=l j=l
aijxij < c Vi;
s.t.
(1)
j=(1)
n
i=l
xij = qj
Vj;
Xij Žo V(i, j).
Note that the complete specification of the allocation mechanism must also include a selection rule discriminating among multiple optimal solutions to AE[b(t)] when needed. Although we do not impose in general any particular selection rule, the convergence results
presented in §3.2 require the optimal solution x(t) to be an extreme point of the feasible
set of AE[b(t)] (which seems both a desirable feature for the buyer, because solutions corresponding to extreme points tend to involve a smaller number of suppliers, and is easy to
achieve in practice, since some optimization algorithms such as the simplex method naturally
generate extreme point solutions). Finally, the allocation engine AE[.] along with this optima selection rule define a function which unambiguously associates a potential allocation
x(t) = (xl(t),...,xn(t)) to any given set of bids b(t), which we can summarize with the
notation x(t) = AE[b(t)].
To encourage competition among suppliers across bidding rounds, our proposed mechanism displays on each supplier i's screen at the end of each round t his own private potential
allocation xi(t) = (xil(t),
2 (t),... ,xim(t));
we refer to this process as private potential al-
location feedback. Each supplier i can then respond to this competitive information in the
following round by submitting a new set of bids bi(t+1), and may in particular lower his bids
on the component types for which his potential allocationxi (t) is deemed unsatisfactory. We
defer until §2.4 a more complete description of the information structure in this mechanism.
11
The relevance of the private potential allocation feedback as a competitive signal (and
hence its ability to generate bidding competition) follows from the activity-based termination
rule that we study. More precisely, the bidding round iterations described above terminate
after the first round T > 1 where no supplier modifies any of his bids from the previous
round, i.e., when b(T) = b(T - 1). At this point in time, the allocation xi(T) calculated
for each supplier i by the allocation engine becomes final (hence the adjective "potential"
used to qualify x(t) for t < T, since the termination round of the auction is not known a
priori), and the final unit prices paid by the buyer to supplier i for the component types
for which he receives a positive allocation are given by bi(T), so that the buyer's final
E bij(T)xij(T).
i=1 j=1
procurement cost resulting from the auction is E
Note that this objective
function does not take into account the fixed cost of establishing and managing a vendor
relationship, so that nothing prevents this auction mechanism from selecting a large number
of suppliers and generating many fragmented orders for the same component type in the
final allocation. More generally, while the stylized auction mechanism just described is well
suited to an analytical study, the allocation environment it captures may be too simplistic
in practical situations. Some possible generalizations, including a bid adjustment system
allowing the buyer to incorporate relative preferences among suppliers (arising, for example,
from incumbent switching costs), are briefly discussed in §4.
2.3. Bidders' Payoff Function.
Our key assumption here is that each supplier i has
a fixed unit production cost vij for each component type j, so that his payoff (or utility)
function in this market mechanism is described byIIi(b(T))
=Y (bij(T) - vij) xij(T), where
j=1
T is the terminal bidding round as previously defined, and (bij(T), xij(T)) are related through
x(T) = AE[b(T)]. This expression can be interpreted as the sum of supplier i's final profits
12
from each component type, in that bi (T) - vi represents supplier i's final unit margin on
component type j, and xi(T) is his corresponding final allocation.
Note first that there is no need to model capacity constraints via overload costs in Hi(),
because by construction all allocations generated by our mechanism will satisfy these constraints. More fundamentally, this payoff formulation implies a linear cost structure, where
for all suppliers the unit production cost of each component type is independent of the quantity allocated. Consequently, our model does not capture the possible economies of scale that
suppliers may experience when they receive large orders (set-up times, production learning
curve, etc.). Nor does it capture the possible diseconomies of scale associated with, for example, the necessary purchase of additional production capacity. While this cost linearity is
a strong assumption, we observe that it is consistent with the auction mechanism we propose to study, where the bids correspond to binding offers to sell at a specified unit price
independent of the quantity received (see §2.2): Because capacity constraints are exogenous
and will always be satisfied in this setting, the reason why suppliers may not be willing to
submit such quantity-independent bids in some situations is precisely because their production costs may not be linear; see Gallien (2000) for a discussion on the design of improved
smart auction mechanisms allowing suppliers to take (dis)economies of scales into account
when they formulate their bids.
Finally, neither our payoff function nor our allocation mechanism capture the possible
synergies (or incompatibilities) across component types, since each supplier's production cost
for one component type is also independent of his allocation of the other component types.
In procurement situations in which such (dis)economies of scope are particularly important,
we believe that some alternate allocation mechanisms (e.g., a corribinatorial auction) may
13
be more appropriate (see references in §1.2).
2.4. Information Structure.
In this section we describe the information structure
underlying our proposed market mechanism, namely which entity knows what information.
We also show how the suppliers' payoff function presented in the previous section can be used
to complete the competitive information feedback provided to bidders during the auction and
lead them to make more efficient bidding decisions.
There are three types of information depositories to consider: The suppliers, the buyer and
the mechanism/auctioneer. We assume here that the auctioneer is trusted by the suppliers to
be respectful of its confidentiality commitments. This is legitimized in our view by the current
fierce competition between online procurement auctioneers (e.g., Freemarkets and eBreviate),
where reputations for integrity standards seem to play a particularly important role: Even
though procurement auctioneers primarily serving buyers are by no means neutral, their
confidentiality commitment seems credible in that the long-term costs of losing the suppliers'
trust will likely outweigh the short-term benefits of communicating to the buyer some private
information obtained from suppliers under a non-disclosure agreement. This along with the
technological medium used to support this auction (a distributed network of computers)
effectively justify the decoupling between the buyer and the mechanism/auctioneer that is
inherent in our mechanism. For studies of competition among auctioneers, see Burguet and
S6kovics (1999) and references therein.
We first discuss the information structure relative to the market environment. Because
the quantities (qj)jEl,...,m} of each component type required by the buyer motivate in large
part the suppliers' interest in the auction, it seems natural that they should be common
knowledge to all participants. However, a key feature enabled by the use of a computer
14
network to run the auction is that the suppliers need not know how many competitors they
have, let alone their identities (for studies of traditional auctions with unknown or uncertain
number of bidders, see Harstad et al. 1990 and references therein). Because this feature
offers some protection against possible collusive behaviors, we have designed our auction
mechanism to take advantage of it. In adherence to this design philosophy, suppliers are not
informed by the mechanism of their competitors' capacity constraints ((aij)jEl,..,m,
Ci)
either. However, this capacity information is required by the mechanism to ensure that
all potential allocations generated satisfy these constraints (see §3.3.4 for a discussion of
the incentive compatibility issues associated with the revelation of their capacity by the
suppliers).
The only dynamic competitive information feedback proposed so far is the private potential allocation feedback x(t), communicated to each supplier i during the auction, but
hidden to his competitors. In particular, because we seek to limit the possibilities of collusion and strategic behavior, our mechanism does not send any direct information to suppliers
during the auction about their competitors' bids (Cramton 1998). Using private potential
allocation feedback alone, however, would probably lead to an inefficient price formation
process, because suppliers would lack sufficient information to rationally decide on which
component types to lower their bids, and by how much. As a result, suppliers in this situation would most likely engage in exploratory bidding behaviors, whereby they would try to
identify the minimal bid decrease leading to some improvement of their potential allocation
through multiple rounds of minimlm bid decrement trials.
The option that we have investigated to prevent this involves providing to each supplier in
every round the possibility to have his myopic best response bid calculated by a local bidding
15
suggestion device. In words, this suggested bid to supplier i is the one that maximizes his
potential payoff function in the next round, assuming that all his competitors' bids remain
unchanged from the current round; i.e., b;(t + l)
sup arg max
II(w, b_i(t))]2 where
m
Lw(eN)
fII(-) is the payoff function defined in §2.3, w represents the decision variables associated
with the bids of supplier i in the following round, and bi(t) is the standard game-theoretic
notation for the bids of all supplier i's competitors in the current round. Note that the
arg max operator acts upon a finite set because of both the non-reneging and the minimum
decrement rules, so that the output of this operator is always non-empty even though II,(.)
is not necessarily continuous. Moreover, the sup operation over the arg max set, provided
it is associated with a total order over (eN) m, ensures that the myopic best response bid is
uniquely defined. With the lexicographic order, for example, suppliers will not be suggested
to lower their bids unless their prospective potential payoff when doing so is strictly greater
than their current potential payoff.
A key information requirement of the myopic best response calculation is that each supplier i willing to use this feature needs to provide the value of his own production costs
(vij),jE(....m} via the auction software interface. Because the production costs of each supplier are sensitive and private information, it is essential for the auctioneer to credibly commit
that these costs, if entered into the myopic best response calculation device, will not be disclosed to the buyer or to any other supplier. We also observe that in end-consumer online
auction sites (Yahoo!, eBay, etc.), users routinely enter their maximum bid for an item into
some software agent (known as proxy bidding, bidding agent, bidding elf, etc.) that then
submits bids on their behalf whenever needed and up to this specified limit (Lucking-Reiley
1999). Because these maximum bid limits are the exact analogue of the production costs
16
Information Available to:
Notation
Information
number and identity
of competing suppliers
component- quantities
required
capacity constraint
of supplier i
production cost
of supplier i
c
(aijq)
Supplier Other
Supplier i
Suppliers
MechanismAuctioneer
Buyer
no
no
yes
yes
yes
yes
yes
yes
yes
no
yes
yes
yes
no
yes
no
yes
no
yes
yes
yes
no
yes
yes
yes
no
yes
no
toj
bids
of supplier i
potential allocation
of supplier i
best response suggestion
of supplier i
hi(t)'
(t)
bz(t)
Table 1: Summary of the information structure.
in our procurement auction setting (and more generally correspond to bidders' valuations
in the traditional auction theory literature), we feel that the myopic best response bidding
suggestion feature is likely to be implementable in practice.
To conclude this section, we summarize in Table 1 the information structure underlying
our proposed auction mechanism. We are now ready to turn to the analysis of our proposed
market mechanism.
3. Analysis
3.1. Methodology.
The basic premise underlying our methodology is that bidders will
be less-than-fully rational when confronted with the smart auction mechanism described
earlier. While from a practical standpoint this assumption simplifies the analysis, we share
with other researchers (Chapter 1 in Samuelson 1997) the belief that it is also motivated
17
by behavioral considerations. In this dynamic game of incomplete information, the classical
solution concepts associated with the requirement of full rationality predict that bidders will
form probabilistic beliefs about all relevant uncertain information in the game, update these
beliefs in a consistent manner during the auction, and derive and execute in every round
a bidding strategy maximizing the expected value of their final payoff function accordingly
(Chapter 8 in Fudenberg and Tirole 1996 and references therein). Because the uncertain
information includes here not only the competitors' production costs, but also their capacity
constraints, their bids and even their number, a rational Bayesian expected utility maximizer
would have to solve an overwhelmingly complex stochastic dynamic program with imperfect
information. The alternate methodology we have adopted consists of assuming a deterministic, less-than-fully rational behavioral model, ultimately specifying for each supplier a
mechanistic relation between the information received in each round and his subsequent bidding decision. This type of model is known in the literature as a nonequilibrium adjustment
or tatonnement process, and is the starting point to the theory of learning in games (see §1.2
for background references).
3.2. Weak Behavioral Assumption and Convergence Bounds.
To establish some
convergence bounds, we only rely initially on the weak behavioralassumption that a supplier
receiving in some round a private allocation feedback of zero for all component types will
lower at least one of his bids, unless doing so would generate a negative profit. Formally,
this means that if xi(t) = 0 and there exists j E {1, ... , m} such that bij(t) > vij + e, then
there exists j E {1,... m} such that bij(t + 1) < bij(t) - e.
We begin by stating an e post result (i.e., relying on an observation of the final allocation made a posteriori) that relates the final winning bids and the order statistics of the
18
production costs.
Let T be the final round of the auction, let v:,,..n,
V:,n
denote the order statistics of (vii,. , v,nj), j E {1,. ., m}, and define PC _ {i E 1, ... ,
xi(T) = O}. Provided that IPCI > 1 and under the weak behavioral assumption, we have
Proposition 1 (ex post bound)
ij (T) >
<bij(T)
vinpJcJ+1: + e Vj E {1,..., m}.
(2)
The proof of Proposition 1 (found in the Appendix along with all remaining proofs) exploits linear programming duality and complementary slackness; we strongly suspect that it
can be extended to a more general class of optimization-based auctions. In words, Proposition 1 states that the highest selected bid on each component type at the end of the auction
cannot be larger than some order statistic of the production costs for that component type
(modulo the minimum bid decrement e), and the greater the number of rejected suppliers,
the lower the rank of this order statistic. Moreover, it seems natural that the upper bound
in (2) involves the number of suppliers eventually receiving no allocation, because the only
behavioral assumption required to derive this result concerns precisely the suppliers receiving a null private allocation feedback. Proposition 1 generalizes to the case of a multi-item
and optimization-based procurement auction the well-known result (Vickrey 1961) that in
a single-item English ascending auction, the price eventually obtained by the seller is equal
to the second highest valuation among the bidders (modulo the minimum bid increment).
Although the information structure is slightly different in our mechanism, the set of parameters m = 1, q1 = 1, ci unbounded and aij = 1 for all i and j, when the allocation selected
is always an extreme point, corresponds to the classical auction of a single indivisible object
with no capacity constraints. In this case, there are IPCl = n - 1 losing suppliers, so that
the bound provided by Proposition 1 is indeed v, + e.
Although Proposition 1 is an e post result, it can be used to prove an ex ante result
19
(i.e., applicable before the auction outcome is observed) in the case where the suppliers have
the same production technology, i.e., all capacity constraints satisfy aij = ai,j = aj V(i, i') E
{1,..., n}. To state this result in a compact form, we first assume without loss of generality
that the suppliers are numbered by increasing order of capacity, i.e., cl < c 2 < ... < c.
This assumption allows us to conveniently define the maximal load number as the smallest
integer p such that
-+1 c, > E ajqj. Note that this definition is not appropriate when
j=1
n
Eci
i=1
m
<
ajqj, but this case can be dismissed because it corresponds to either infeasible
j=1
buyer requirements or, at equality, to the trivial situation in which all the available supply
capacity is required. The observation that the maximal load number provides an upper
bound on the number of suppliers loaded to their capacity in any feasible allocation is key
to the following proposition.
Proposition 2 (ex ante bound) Assume that the optimal solution selected by the allocation engine AE[-] is always an extreme point, that all the suppliers have the same production
technology, and that the maximal load number p satisfies n - p > m + 1. Then under the
weak behavioral assumption, we have
xij(T) > O= bi (T) < v+m+l:, +
(3)
C,
and the buyer's final procurement cost is bounded by
bi
i=1 j=1
(T)x 2ij (T) < Eqj
(v
+l:n+
.
(4)
j=1
This last result is noteworthy in that it provides a performance guarantee for the buyer
before the auction takes place, even though the behavioral assumptions required are particularly mild. As in (2), the upper bound (4) generalizes in the setting of a multi-dimensional
optimization-based procurement auction the relation between winning bids and valuations
in simpler auctions. However, it shows much more explicitly how this bound for the final
bids depends on the primary data of the problem, namely the suppliers' capacity constraints
and production costs, and the buyer's requirements. In particular, the larger the buyer's
20
required quantities in relation to the suppliers' available capacities, the looser the bound
on the final bids; hence, this expression provides quantitative support for the intuition that
the buyer's market power decreases as supply resources become more scarce. Likewise, the
applicability condition n - p > m + 1 can be interpreted as a minimum requirement on
the level of competition in the auction (via the number of participating suppliers) or on the
buyer's market power. Finally, the upper bound (4) helps reveal the relationship between
the buyer's final procurement cost and her required quantities; note that in this model with
linear production costs, an increase in the component market size may potentially increase
the cost per unit because of the competitive effects.
3.3. Myopic Best Response. We now introduce a fully specified behavioral model, which
follows directly from the myopic best response (MBR) information feedback described in §2.4.
Namely, we assume that every supplier uses in each round the MBR calculation device and
follows exactly its bidding suggestions, i.e., bi(t + 1) = b;(t + 1) for all i E {1,..., n} and
all t > 0, where b(t + 1) is the expression defined in §2.4. Note that the MBR bidding
model almost satisfies the weak behavioral assumption, the exception being the suppliers
with no current potential allocation who may still not benefit from lowering any of their bids
(e.g., when all competing bids are lower than his production costs), because in such cases
the sup operator used in the definition of b(-) will cause these suppliers' bids to remain
constant. Nevertheless, the convergence bounds established in §3.2 remain valid under the
MBR behavioral model, as can be quickly checked from their respective proofs.
From a mathematical standpoint, the MBR behavioral model amounts to a recursive
relation between consecutive values of the bidding sequence (b(t))tN of the form b(t + 1) =
.F[b(t)], t > 0. Because F[ *]is a well-defined function (see §2.4), the entire bidding history
21
of the auction is completely characterized by the initial bids b(O) and this recursive relation.
A property of the MBR bidding sequences essential to the theoretical validity of our solution
concept is that they always converge in finite time (they are non-decreasing by the nonreneging rule, bounded from below and may only take a finite set of values by the common
multiple rule). This can be formalized by the following proposition.
Proposition 3 Let (b(t))tEN be a myopic best response bidding sequence defined by a set of
initial bids b(O) and the recursive relation b(t + 1) = F[b(t)]. There exists an integer T > 0
such that b(t) = b(T) for all t > T.
By the recursive definition of the MBR bidding sequence (b(t))tEN, its limit b(T) is
necessarily a fixed point of the relation
.F[-],
i.e., b(T) = F[b(T)]. Conversely, for every
fixed point b of F[.], there obviously exists a MBR bidding sequence converging to b (e.g.,
b(O) = b). The solution concept we propose to predict the final outcome in our market
mechanism given an initial set of bids b(O) is precisely the limit b(T) of the MBR bidding
sequence (b(t))tEN. This is the natural adaptation to our setting of the auction-specific
local Nash equilibrium defined in Bykowsky et al. (1995) in the context of the simultaneous
ascending auction.
3.3.1. Model Discussion. Although the MBR behavioral model is close to the classical
Cournot or best response tatonnement process (Chapter 1 in Fudenberg and Tirole 1996),
there remains a few important differences between them. First, the information structure of
the game to which they apply differ, since the duopoly competition studied by Cournot is
a game of common knowledge/full information, whereas our auction format corresponds to
a game under uncertainty. Moreover, payoffs in our mechanism are not obtained at the end
of every round as in the Cournot game, but only at the end of the auction. Consequently,
the objective function used in each supplier's MBR calculation only represents his potential
22
payoff in the next round, and only coincides with the actual payoff in the round immediately
preceding the end of the auction. The MBR behavioral model thus implicitly assumes the
belief that the auction will terminate in the next round (see discussion two paragraphs
below on why this is still a reasonable assumption in this setting), justifying the use of the
adjective "myopic". Finally, because of the non-reneging bidding rule, decision spaces in
different rounds are not independent from each other as in the Cournot adjustment. More
precisely, each bid constrains the future bidding decisions of the supplier who submitted it,
and this constraint is more stringent as the value of this bid is lowered.
These specific features provide what we feel is a strong justification for applying the
MBR model to the game under study. In the classical case with no uncertainty, the best
response adjustment process has been sharply criticized for the players' naiveness and/or
reasoning inconsistencies it implies: While it relies in every round on the assumption that
the competitors' actions will remain constant in the next round (strategy stability), this very
assumption is observably proven wrong in every round until equilibrium is reached (Fudenberg and Levine 1998). However, in the information structure underlying the game we study,
bidders cannot observe their competitors' bids directly, and the indirect process by which
they could try to infer these bids - based on the history of their own bids, private allocation
feedback and MBR bidding suggestions - seems particularly complex. As a consequence,
the assumption of strategy stability is plausible, in sharp contrast to more classical games
where past actions are common knowledge.
More fundamentally, suppliers in this auction will be particularly cautious not to submit
low bids prematurely, because of the lock-in effect induced by the non-reneging rule. In this
setting, strategy stability is likely to be perceived as a wise assumption on which to base
23
bidding decisions, even when also perceived as probably wrong. This is because incorrectly
assuming that the competition will be more aggressive in the next round may lead to lower
bids than necessary, thus wasting some margin and restricting the space of future possible
actions. On the other hand, the activity-based termination rule ensures that a supplier
who underestimates his competitors' aggressiveness in a given round by assuming strategy
stability will not be penalized: If this supplier ends up lowering one of his current bids then
the auction will continue for at least one more round, giving him the opportunity to adapt
his bids to the actual bidding strategy of his competitors. If this supplier does not modify
any of his bids then the auction may terminate, proving the strategy stability assumption
correct and therefore his bidding decision optimal, or it may continue, leaving him the option
to further modify his bids.
3.3.2. Impact of Initial Bids. Although Propositions 1 and 2 show that the dependence
of the auction equilibrium on the initial bids should be relatively limited when the level
of competition among suppliers is sufficiently high, these results do not apply in market
environments characterized by supply resource scarcity. To gain some insights into the
impact of initial bids when the level of competition is low, we turn to the analysis of the MBR
bidding dynamics in the simplest non-trivial market environment in which our smart-market
mechanism can be applied: The 2 x 2 symmetric case, where n = 2, m = 2, ql = q2
=
q,
aij = 1 (i, j) and cl = c2 = c, but vl need not equal v 2. The only interesting choices of
parameters in this environment satisfy 1 < 8 < 2, where 6 = k, which is the number of
suppliers required to cover the total procurement needs'. For this range of 6, the buyer has
no choice but to use a positive amount of capacity from both suppliers.
1 In the case 8 < 1, the allocation of each component type is entirely independent from the other, and the case8 >
2 is infeasible.
24
The following proposition shows that even with such scarce supply capacity, when the
initial bids are sufficiently close to each other and are greater than the component-wise
maximum of the production costs, they have relatively little impact on the final outcome.
Moreover, it also shows that the particular information structure used in our mechanism
may still enable some bidding competition.
Proposition 4 Let d(b 1 , b 2 ) = max(lb 2l - b1 ll, b22 - b12 j). In the symmetric 2 x 2 auction and under the MBR bidding model, there exists a selection rule discriminatingbetween
multiple optimal solutions to the allocation engine AE[.] such that
d(bl(0),b2(0)) < e =* d(bi(T),vl V 2) < (
3q - c
c-q
+ 1)e, i
{1, 2}.
A legitimate question at this point is to determine to what extent the final outcome is
affected when the initial bids are further apart. Although we do not entirely resolve this
issue here, it is revealing to plot for a particular choice of parameters (c, q, vi, b 1 (0)) the
stability zones for supplier 1, i.e., the subset of supplier 2's bidding space such that supplier
1 would not move in the next round should b 2 (0) belong to it (see Figure 1).
Figure 1 shows the existence of premature equilibria, arising when the initial bids are too
far apart, so that the increased order volume the higher bidder may receive by outbidding
his competitor does not compensate for the margin reduction that would be incurred. For
example, if supplier 2's bid was to lie in the lower-left quadrant of the stability zone, supplier
1 would not move in the following round by definition. Depending on supplier 2's production
costs, he will either not move (thereby achieving a premature equilibrium) or may try to
switch the component type for which he received the larger volume. If these observations also
apply to environments with more suppliers and component types, then there are implications
for mechanism design: Whenever supply capacity is suspected to be scarce, it seems in the
buyer's interest to have initial bids as close to each other as possible, which could be achieved,
25
Bids/costs on
comp
Bids/costs on
_,,s^^,f,
Figure 1: Supplier l's stability zones (shaded areas).
for example, by enforcing a common minimum entry bid.
3.3.3. Bidding Dynamics.
When the initial bids are sufficiently close, the proof of
Proposition 4 also reveals the structure of the bidding dynamics in the symmetric 2 x 2 auction
according to the MBR model. This structure is illustrated in Figure 2, which represents a
typical MBR bidding sequence. Here, the initial bids are below the 45 ° lines going through
the production costs of both suppliers, which we refer to as the margin switching lines
for a reason that will soon become clear. As long as the bids remain in this area of the
bidding plane, both suppliers try to outbid their competitor primarily on component type
1 (so as to receive the full order quantity of this component), because it is the one with
the higher relative margin. Thus, the bidding sequence moves horizontally, until it hits
supplier 2's margin switching line, at which point supplier 2 starts competing primarily
for component type 2. The bidding sequence then moves downwards along this line, until
26
it hits the horizontal line going through supplier l's cost for component type 2, at which
point this supplier stops competing altogether on this component type (his margin is then
reduced to zero). The bidding sequence starts moving horizontally again, before it reaches
an equilibrium in the neighborhood of vl Vv 2 , as predicted by Proposition 4.
Rid-,ctw
n
bi(O)
(starting bid of
supplier 1)
rUngbid of
plier 2)
(product
costs of
supplier
ids/costs on
oreponent 1
Figure 2: Bidding dynamics in the symmetric 2 x 2 auction.
Because of the combinatorial complexity involved, it seems hard to perform a similar analytical study of the MBR bidding dynamics in market environments with more suppliers
and/or more component types. Instead, we undertook some numerical experiments simulating competitive interactions under the MBR behavioral model. The two experiments with
eight suppliers and two component types represented in Figures 3 and 4 are fairly typical
of the results we have obtained (see Gallien 2000 for more examples). These figures show
each supplier's bidding sequence in the auction as a color-coded dotted line (a color version
of these figures are available from the authors), and the corresponding production costs as
diamonds. In addition, the set of connected squares is the sequence of dual prices associated
27
in each round with the quantity requirement constraints in the formulation of the allocation
engine AE[.] (referred to below as market prices according to the classical interpretation
of LP duality theory). By design, these two experiments have the exact same parameters
(capacity, required quantities, initial bids), with the exception of the production costs, which
are more homogeneous across component types for the experiment 2 reported in Figure 4.
In Figure 3, all suppliers except one exhibit a specialized pattern of bidding, where they
compete by decreasing their bids for only one of the two components. However, in Figure
4, there are many more "diagonal" bidding paths, i.e., competition takes place for more
suppliers along the two component axes. The primary driving force behind these different
bidding patterns seems to be the relative position of each supplier's production costs in each
round with respect to the market prices. More specifically, as the auction evolves, the path
formed by these market prices partitions the suppliers' production costs into three categories.
Suppliers with production costs roughly proportional to the market prices tend to compete
for both components, while suppliers with production costs clearly above or below the path
of market prices typically only compete for the one component on which their current relative
margin is the highest (as defined by the relative position of their production costs and the
current market prices). This interpretation explains why more homogeneous production
costs across components (as in Figure 4) lead to a competition pattern that is more mixed.
3.3.4. Incentive Compatibility.
Although the incentive compatibility of suppliers'
production costs input when using the MBR bid suggestion device is a legitimate question in
theory, experiments described in Gallien (2000) tend to show that a truthful cost revelation
is compatible with suppliers' incentives, so that we do not discuss this issue in further detail
2
For each supplier, production costs in Figure 4 have been obtained from those of Figure 3 by a translation along
the off-diagonal.
28
25
20
115
8
710
Sm
5
0
15
10
20
25
BidaiCosts on Compont 1
Figure 3: MBR dynamics in 8 x 2 auction with heterogeneous costs.
here. Instead, we focus on the incentive compatibility issues linked to capacity revelation.
A simple-minded analytical approach to this problem is to ignore the potential impact
of a supplier's capacity input on the other suppliers' bidding behavior, and to perform a
sensitivity analysis on this capacity input. That is, assuming that the bidding sequence has
reached in round T an equilibrium b(T), what is the impact of a marginal change of supplier
i's capacity cj on his payoff function IIi(b(T)) =
[bij(T) - vij] xii(T)? By complementarj=-
ity slackness, the dual variable associated with the capacity constraint of supplier i (which
represents the marginal change in the buyer's objective resulting from a change in ci) can
only be positive if E xij(T) = ci. Hence, a necessary condition for this marginal change to
j=1
have any impact on supplier i's allocation is that his capacity constraint be binding. That
is, a supplier cannot benefit by overstating his capacity without becoming overloaded and
29
25
20
c 15
E
a
a
C)
0
Xu
10
m
5
0
0
5
10
15
20
25
Bids on Component 1
Figure 4: MBR dynamics in 8 x 2 auction with homogeneous costs.
incurring an observable default (i.e., refusing to accept all of the work he has been awarded).
However, this static sensitivity analysis does not take into account the strategic implications of a capacity statement on the other suppliers' bidding behavior. More specifically, it
is conceivable that a supplier could earn higher margins by understating his capacity, because it may induce the other suppliers to bid less aggressively, which could compensate for
the lower market share resulting from the capacity understatement. Likewise, this strategic
effect could play against the incentive to overstate one's capacity, because it would induce
the other suppliers to bid more aggressively.
Although it appears difficult to assess these strategic effects analytically, numerical experiments conducted under the MBR behavioral model and reported in Gallien (2000). confirm
the abovementioned hypothesis that benefits derived from overstating one's capacity neces-
30
sarily imply overloading (as opposed to strategic implications on other suppliers' behaviors).
We also report in Gallien (2000) some "cooked-up" experiments demonstrating that it is
possible for a supplier to increase his profits by understating his capacity (which can also
be interpreted as a disincentive for capacity overstatement), thus illustrating the strategic
effect hypothesized earlier.
In summary, it appears that there is no incentive for a supplier to overstate his capacity,
and there are examples, which require more information (e.g., the other suppliers' capacity constraints), rationality and risk-taking than would typically be observed in practice,
in which a supplier can raise his profits by understating his capacity. Hence, if suppliers
are penalized for capacity overloading, then our smart-market mechanism should be able to
induce truthful revelations of suppliers' production capacity. Regarding penalties, it is interesting to note that Freemarkets keeps instances of capacity overloading to a minimum by
suspending from subsequent bidding events overloaded suppliers who default after an auction
(Wnorowski 2000); it is this punitive action that led to the observed impact of overload fear
on suppliers' bidding behavior, which in turn motivated the design of our smart-market approach (see §1). Alternatively, a procurement contract specifying an appropriately high fixed
penalty in the event of misdeliveries and/or quality problems may also ensure a relatively
truthful capacity input by the suppliers.
4. Conclusion
Partly motivated by the observed inefficiencies of current online procurement auction mechanisms when suppliers' capacity constraints are stringent, this paper investigates an alternative mechanism designed specifically for such environments. It relies on an estimation of these
31
capacity constraints prior to the bidding event, which enables the use of an optimizationbased allocation engine (justifying the term "smart market"). For every set of bids submitted
during the auction, this engine dynamically computes the allocation of procurement contracts
minimizing the buyer's total cost under both the buyer's quantity requirement and the suppliers' capacity constraints. The bidders' information sets in this mechanism are designed
to achieve a middle ground between facilitating competition and preventing collusion, exploiting some of the possibilities offered by the use of a distributed network of computers
to support the auction. They include in particular an original myopic best response bidding
suggestion device, taking as an input a supplier's production costs, and delivering as an output the set of bids maximizing his potential profit in the next round under the assumption
that his competitors' bids remain constant.
The solution concept we propose for this auction mechanism is based on tatonnement
theory, whereby we first formulate less-than-fully rational behavioral models for the bidders,
and then examine the implications of these models on the dynamics and the convergence
properties of the resulting bidding sequences. Under a weak behavioral assumption loosely
characterizing the reactions of potentially rejected suppliers, some symmetry requirements
and a minimum competition level condition, we constructed upper bounds for the bids of
the winning suppliers at the end of the auction, providing in turn an upper bound on the
buyer's final procurement cost. These bounds were shown to carry over to the complete
behavioral model, in which the suppliers followed the myopic best response bid suggestions.
We then derived some additional structural properties of the myopic best response bidding
sequences in a symmetric (except for production costs) market with two bidders and two
component types, which suggested that when supplier capacity is scarce (i.e., competition
32
is low), the buyer may find it beneficial to impose a common maximum entry bid to discourage premature equilibria that are generated by initial bids that are far apart. Finally,
by performing numerical simulations of myopic best response sequences in more complex
market configurations, we inferred that the negative incentive compatibility effects linked
to the revelation by the suppliers of their private capacity and cost information should be
relatively limited.
We believe this work constitutes a useful basic framework towards the development of
electronic trade systems enabling real-time complex industrial transactions. Within the
scope of our model, natural steps in this direction include more complex capacity constraints,
nonlinear production costs and bid structures, requirements on the supply base size, and
incumbent switching costs; this last generalization is particularly important, because these
switching costs (which could, more generally, incorporate the bidder's perceptions of the
suppliers' reputations) are the primary reason why Freemarkets makes its allocation decisions
after - rather than during - their bidding events. In the framework of our model, one way for
the buyer to account for her relative preferences across suppliers is to use a bid adjustment
mechanism. In this method, the actual bids bii submitted by suppliers are automatically
adjusted for the purpose of computing potential allocations by an additive factor cij > 0
reflecting the buyer's relative preferences. That is, while the selling prices resulting from
the auction still correspond to the actual bids bij submitted by the winners, the potential
and final allocations of components are now calculated by solving x(t) = AE[b(t) - c]; the
higher acij, the more supplier i is advantaged relative to his competitors for the allocation of
component type j. Some of our results can be generalized to include these adjusted bids. For
example, the right side of the ex-ante bound in Proposition 2 becomes [v - ctjm+pl:n + E+
33
aij, where [v - a]!j
denotes the k-th order statistic of (vij- aj,...
,v
- aj). In words,
modifying the bids for allocation purposes through the mechanism just described roughly
amounts to modying the production costs by the same adjustment factors, as one would
intuitively expect.
Unfortunately, this adjustment model does not allow us to capture fixed switching costs
(i.e. independent of the quantity allocated). From a practical standpoint, this particular
extension along with most of the others previously mentioned can be incorporated by using
more sophisticated integer programming formulations for the allocation engine AE[-] in (1)
(see Gallien 2000 for details), although we have not derived theoretical results corresponding
to Propositions 1-4 in these more complex settings. Another relevant challenge would be
to model into our allocation engine more criteria specific to industrial procurement, such as
delivery performance, quality, and insurance provisions, leading perhaps to the design of a
multi-dimensional bidding mechanism. In such complex and dynamic trading environments,
efficiency will require the design of decision support tools available to bidders, for example
by generalizing our myopic best response bid computational device. From a methodological perspective, we note that formulating behavioral models associated with these decision
support tools seems to be a fruitful approach when trying to predict various mechanism
outcomes.
Appendix
A.1. Proof of Proposition 1.
Let z(T) denote the value in round T of the optimal dual
variables associated with the quantity constraints in AE[b(T)I, and e, an m-dimensional
34
vector with all its components equal to 1. Then we have
i E pC
vi > z(T)- eem.
(5)
This is because bi(T) > z(T) for any supplier i E PC, so that if we had vij < zj(T) - e
for some j, then supplier i could increase his profit (currently equal to zero) by bidding
bij(T + 1) = zj(T) - e < bij(T). But bi(T + 1) = bij(T) by definition of T, and therefore
zj(T) - e < vij Vj. This last statement applies to at least IPCI suppliers, so that
j(T) <
p
+ .
(6)
Finally, xij (T) is the dual variable associated with the constraint zj (T) < bij(T) + aijyi of the
dual to AE[b(T)], where yi > 0 is the dual variable associated with the capacity constraint
of supplier i in AE[b(T)]. Therefore, xij(T) > 0 implies bij(T) < zj(T), which combined
with (6) completes the proof.
A.2. Proof of Proposition 2.
The proof amounts to finding a lower bound for pC[
(or equivalently, an upper bound for PI), and then invoking Proposition 1 in the cases
where this bound allows us to claim that pc[ > 1. In any feasible allocation, the number of
suppliers who are fully loaded is clearly smaller than the maximal load factorp, because of
the quantity requirement in (1). We now claim that in any extreme point of the polytope of
feasible solutions to (1), the number of suppliers that are not fully loaded but still receive a
positive allocation is smaller than m. This is because the allocation would otherwise include
at least two non-fully loaded suppliers receiving a positive allocation of the same product,
and would therefore not be an extreme point (as it could be written as a convex combination
with positive weights of two distinct feasible solutions). Therefore, among all the suppliers
35
receiving a positive allocation of each product, at most one can be non-fully loaded, and
the total number of non-fully loaded suppliers is smaller than m. This completes the proof
of IPI < p + m, from which (3) follows immediately by Proposition 1. The bound on the
buyer's final procurement cost is a direct consequence of (3).
A.3. Proof of Proposition 4.
A.3.1. Notation.
In this market environment, AE[.] has six extreme points, namely
(x 1,x 2) E {(A, C), (C,A), (B,B), (B, B), (A, C), (C,A)}, where A = (2q - c, )T, B =
(q, O)T, C = (q, c - q)T
(i.e., A = (0, 2q-
and ~ (tilde) denotes an exchange of the two elements in the vector
c)T ).
In this proof, we refer to h(t) and h(t + 1) as h and h, respectively,
where h denotes any variable or vector of interest (allocation, bids). Also, to describe a
player's strategy for the next round, we use the notation by where Y E {A, B, C, A, B, C}
with Y = (AE[bi = by, bi])i (i.e. under the MBR assumption that b_i = b_i, player i will
obtain an allocation i = Y if he plays by). As shown in Gallien (2000), for a given selection
rule, by can be uniquely defined among all bids yielding allocation Y under strategy profile
stability as the one requiring the smallest decrease in margins (so that each player's strategy
space at each round is practically discrete). When comparing the impact of two different
bid choices on a player's payoff function, we refer to the case II(bi = by, bi) > IIi(bi =
bz, b-i) as by > bz. Finally, we use in the bidding space the distance metric defined by
d(bl, b 2 ) = max(b 2l - blll, b22 - bl21)
A.3.2. Selection Rule.
We consider a multiple optima selection rule symmetric across
bidders and component types, where the selected allocation is always an extreme point, and
that is characterized by
36
b12 >
*
622
bL >b21
bll-b 2 1
*
-
2
(xl,x2) = (A,);
-= 22
21 = L2-22
bll > b21
* bl = b2 • (l,
(l, x2=
(A, C) if (xl, x 2)
(A, C) otherwise
E
{(C, A), (B, B), (A, C)}
2) = (xl,x 2).
Note that all missing cases in the definition above can be resolved by symmetry, and a
random selection rule can be used at the first round if necessary.
A.3.3. Strategy Space Restrictions.
While at every round the strategy space of
each player is a priori {bA, bB, bc, bA, bA, be}, it can typically be reduced to a smaller set
by considering both the non-reneging rule (e.g., if xl = B then bl E {bB, bc, bc,}) and an
adaptation of Lemma 4 in Part I of Gallien (2000) to the selection rule described earlier,
which shows that under the MBR rationale:
1. If b22 - v12 > b2 - vl (player 2's bid is above player l's margin switching line), then
1
bA > bA;
2. If b22 - v 12 > b2l 1
3. b22 > v12
11 and xl E {A, A, B, C}, then be
> be;
.1
bc > be and b2l > vill
bo > bB.
Note here again that many more such statements follow by symmetry. In summary, the
set of possible bid choices b, for player 1 can be obtained from the following table (in order
to avoid a long and uninteresting discussion of the tie cases, we assume that the production
costs vl and v2 do not belong to the e -grid):
37
-
xl
__
__
A
A
{bA, bo}
{bA,bo}
{bA, bB}
__
B
___
C
B
Case:
b22 - V12 > b21 - v1 1 and b 2 > vl
b2 1 < Vll
b2l < V
b 22 - v 12 > b2 1 - v11 and
{bA, bB}
{bC, bC} {bo}
N/A
{bA}
{bc, bC}
{be}
N/A
N/A
Let us now assume that
A.3.4. Neighborhood Stability and Convergence.
d(b1 ,b 2 ) < e. We can use the above table to specify the bid chosen by player 1 under
the MBR rationale (results for player 2 and for missing cases follow by symmetry):
Case 1: b22 - v1 2 > b2 - v
and b 2 > V1
- When x 1 E {A, A}, we have
1
bA > b
(2q - c)(b 22 + e - V1 2) > q(b 22 - e 2 b22 -
12
+ b21 - V11 <
1 2)
+ (c - q)(b 21 - v)
c-q
c-q
- When xl = B,
bc > bC
¢ (c- q)(22 - v 1 2) + q(b -' V~) > q(522 b22 - 12 - b21 + V11 < 2q-c
-
v 1 2 ) + (c-
q)(b 1 1 - vl)
- When x1 = C,
be > b e
Case 2: b22 -
4 q(b1l - vjl) + (c - q)(bl2 -
i - V12 - bll + vll < 2q- e
V2 > b2 l -vl 1
1 2)
> (c - q)(bl - vll) + q(b l 2 - e -
1 2)
and { b2 < v11
b22 > V1 2
- When xl E {A, A}, we have
1
bA > bB if (2q- c)(b2 2 4 b22 - V12 < c-'q e
1 2)
> q(b 22 - e-
1 2)
In words, supplier 1 prefers bc to bA as long as supplier 2's bid b 2 remains sufficiently
greater than vl (i.e., when b22 - v12 + b2l - V11 > 3e),, and prefers be to bc when b 2 is far
enough above his margin switching line (i.e., when b12 - v2 - bll + v1 >
2q-c)
When b21
drops below v l l , supplier 1 plays bg if b22 is sufficiently greater than v 12 (i.e., when b22-v
12
>
4cLq), and bA otherwise. Applying the same reasoning to player 2, we can construct for all
38
;
relevant cases the tables describing the joint bidding strategies and resulting allocation for
both players, thus characterizing the dynamics of the bidding sequence. For example, the
b2 -
table corresponding to the case
bbf
2 > bl - v
v 2 > b1 -
1
and b2 > Vl
and bi>v2 is (missing allocation
cases can be derived by symmetry)
B:
A:
A:
.
d(b2, vl)
< 3--c
C-q
-
(be, b)
- (A, C)
C: bi2-
c-q
d(b 2, vi)
< 3-c
-C-q
_
_
b22
b22 - V12
-b 21 + Vll
- V12
-b 21 + V1
> 2a-c
C--c
<
2-=ce
2a-c
(bCe bA)
-
(C,A)
-bll + Vii
2g-c
3q-cq
c
c-q
V12
C: b2 - V1 2
-bll + vii
d(b2, vi)
d(b2 , vl)
,
i
(be, b)
(be, bA)
- (c,A)
- (C, A)
(bC, bC)
- (A, C)
(bc, bA)
- (C, A)
I
(be, be)
B
-
(A,C)
(be, bc)
- (B,B)
where the rows correspond to player 2's allocation x 2, the columns to xl, and each entry
(I, J) in this table provides (b 2 , bl) -
(x2, l) when (x 2 ,xl) = (I, J). Constructing the
tables corresponding to cases
>v
i2 -bv1 2 > bll -v2 and bl and
b2
- V22 < b1 l - V21 and bi > -v
, b21 <v11 andb 22 >
V22 > bl - v 2 1 and b > v 2
bl bn -v 1 2 >b
b12 -
12
(tables for other cases follow by symmetry), we can observe that
d(bi, b2) < e;
bi2 -_i2 + bi - v-i > e for i E {1, 2}; and
biij- v-i > qcs for i,j E 1, 2}
implies
d(b 1,2) < e; and
bl1 < bl,b12 < b12 ,b 21 < b21 or b22 < b22
In words, if at one given round the bids of the two suppliers are within a distance e of each
39
other and sufficiently far away from the production costs, then under the MBR rationale
the bids at the next round will still be within e of each other, and at least one of the
bids will have been strictly decreased. Applying this reasoning iteratively, observing that
3q-ce
for i = 1 or 2 implies
c-q
c-q > cqe
c-q since 2q - c > 0, and that bi2 - v_i2 + bil - v-il < 3q-ce
d(bi, vl Vv 2 ) < 3q-e, we can conclude that d(bi(T), vl V v2 ) < (
+ 1)E for i E 1, 2},
which completes the proof.
Acknowledgment
We thank Jim Orlin for valuable suggestions leading to the weakening of some of the initial
assumptions for the convergence results reported in §3, Sam Kinney and Doug Wnorowski for
sharing details about the operations of FreeMarkets, and Yashan Wang, Sridhar Tayur, John
Sterman, Erik Brynjolfsson, Nelson Repenning, Rachel Croson, Robert Porter, Glenn Ellison
and Andreas Schulz for provocative discussions and/or suggestions of relevant references. We
are also grateful to Ozlem Ergun for reviewing some conjectures. This research was supported
in part by the Singapore-MIT Alliance and a Ph.D. fellowship from the MIT Sloan School
of Management.
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_1IP
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